The (semi)stable loci are defined in [1]} using \(G\) -invariant sections of positive powers of \(\mathcal {L}\) . Critical to the use of reductive GIT in practice, though, is the ability to determine these loci without computing ring of invariants, afforded by the Hilbert–Mumford criterion. This criterion states that a closed point \(y \in Y\)  is semistable if and only if it is semistable for each 1-parameter subgroup (1-PS) \(\lambda \)  of \(G\) . To state this as a numerical criterion, we define the Hilbert–Mumford weight \(\mu (y,\lambda )\)  to be minus the weight of the \(\mathbb {G}_m\) -action induced by \(\lambda \)  on the fibre of \(\mathcal {L}\)  over the fixed point \(\lim _{t \rightarrow 0} \lambda (t) \cdot y\) .
